Complexity Analysis of Term Rewrite Systems
Georg Moser
Institute of Computer Science
University of Innsbruck
IFIP WG 1.6, July 2, 2009
Overview
The Fundamentals
The Past
The Present
The Future
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
2/21
The Fundamentals
The Fundamentals
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
3/21
The Fundamentals
Definition
derivation length
n
dl(t, →) = max{n | ∃u t → u}
dl(n, T , →) = max{dl(t, →) | ∃t ∈ T and |t| 6 n}
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
4/21
The Fundamentals
Definition
derivation length
n
dl(t, →) = max{n | ∃u t → u}
dl(n, T , →) = max{dl(t, →) | ∃t ∈ T and |t| 6 n}
Definition
derivational complexity
dcR (n) = dl(n, ”all terms”, →R )
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
4/21
The Fundamentals
Definition
derivation length
n
dl(t, →) = max{n | ∃u t → u}
dl(n, T , →) = max{dl(t, →) | ∃t ∈ T and |t| 6 n}
Definition
derivational complexity
dcR (n) = dl(n, ”all terms”, →R )
Definition
runtime complexity
rcR (n) = dl(n, ”basic terms”, →R )
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
4/21
The Fundamentals
Definition
derivation length
n
dl(t, →) = max{n | ∃u t → u}
dl(n, T , →) = max{dl(t, →) | ∃t ∈ T and |t| 6 n}
Definition
derivational complexity
dcR (n) = dl(n, ”all terms”, →R )
Definition
runtime complexity
rcR (n) = dl(n, ”basic terms”, →R )
term f (t1 , . . . , tn ) is basic if
• f is defined
• t1 , . . . , tn contain no defined symbols
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
4/21
The Fundamentals
How To Analyse Complexity
t1 →R t2 →R t3 →R . . .
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
5/21
The Fundamentals
How To Analyse Complexity
t1 t2 t3 . . . tn
consider
1
∃ reduction order such that
2
R⊆
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
5/21
The Fundamentals
How To Analyse Complexity
t1 →R t2 →R t3 →R . . . →R tn
consider
1
∃ reduction order such that
2
R⊆
Observation
• can be used to measure the derivation length
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
5/21
The Fundamentals
How To Analyse Complexity
t1 →R t2 →R t3 →R . . . →R tn
consider
1
∃ reduction order such that
2
R⊆
Observation
• can be used to measure the derivation length
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
5/21
The Past
The Past
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
6/21
The Past
Complexity in Rewriting: A History
# of papers
RTA, TCS & IC
(1481)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
7/21
The Past
Complexity in Rewriting: A History
# of papers
complexity of TRSs (20)
RTA, TCS & IC
(1481)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
7/21
The Past
Complexity in Rewriting: A History
# of papers
termination (187)
RTA, TCS & IC
(1481)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
7/21
The Past
Complexity in Rewriting: A History
3 selected papers
Algorithmic Complexity
of Term Rewrite Systems;
Choppy et al., RTA 1987
RTA, TCS & IC
average cost analysis
(1481)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
7/21
The Past
Complexity in Rewriting: A History
3 selected papers
Termination Proofs and the
Length of Derivations; Hofbauer, Lautemann, RTA 1989
RTA, TCS & IC
to be continued
(1481)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
7/21
The Past
Complexity in Rewriting: A History
3 selected papers
Termination Proofs by Lexicographic Path Orders imply
Multiply Recursive Derivation
Lengths; Weiermann, TCS
1995
RTA, TCS & IC
(1481)
LPO simple, complexity wise
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
7/21
The Past
Termination Proofs and the Length of Derivations
• introduction of derivation length, derivational complexity
• derivational complexity as measure of a termination technique
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
8/21
The Past
Termination Proofs and the Length of Derivations
• introduction of derivation length, derivational complexity
• derivational complexity as measure of a termination technique
Theorem
Hofbauer, Lautemann
polynomial interpretations induce double-exponential derivational
complexity
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
1989
8/21
The Past
Termination Proofs and the Length of Derivations
• introduction of derivation length, derivational complexity
• derivational complexity as measure of a termination technique
Theorem
Hofbauer, Lautemann
polynomial interpretations induce double-exponential derivational
complexity
1989
Lemma À
∀ R terminating via a polynomial interpretation
c·|s|
∃ c ∈ R, c > 0 ∀ terms s: dl(s, →R ) 6 22
Lemma Á
∃ R terminating via a polynomial interpretation
c·|s|
∃ c ∈ R, c > 0 for infinitely many terms s: dl(s, →R ) > 22
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
8/21
The Past
Proof of Lemma Á
consider Rhl :
x+0 → x
x+s(y ) → s(x+y )
GM (ICS @ UIBK)
d(0) → 0
d(s(x)) → s(s(d(x)))
q(0) → 0
q(s(x)) → q(x) + s(d(x))
IFIP WG 1.6, July 2, 2009
9/21
The Past
Proof of Lemma Á
consider Rhl :
x+0 → x
x+s(y ) → s(x+y )
0A = 2
GM (ICS @ UIBK)
sA (n) = n + 1
d(0) → 0
d(s(x)) → s(s(d(x)))
q(0) → 0
q(s(x)) → q(x) + s(d(x))
n +A m = n + 2m
IFIP WG 1.6, July 2, 2009
dA (n) = 3n
qA (n) = n3
9/21
The Past
Proof of Lemma Á
consider Rhl :
x+0 → x
x+s(y ) → s(x+y )
0A = 2
1
2
3
sA (n) = n + 1
d(0) → 0
d(s(x)) → s(s(d(x)))
q(0) → 0
q(s(x)) → q(x) + s(d(x))
n +A m = n + 2m
dA (n) = 3n
qA (n) = n3
s defines the successor function
∗
d defines the doubling function, i.e., d(sn (0)) →
− s2n (0)
2
∗
q defines the square function, i.e., q(sn (0)) →
− sn (0)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
9/21
The Past
Proof of Lemma Á
consider Rhl :
x+0 → x
x+s(y ) → s(x+y )
0A = 2
1
2
3
sA (n) = n + 1
d(0) → 0
d(s(x)) → s(s(d(x)))
q(0) → 0
q(s(x)) → q(x) + s(d(x))
n +A m = n + 2m
dA (n) = 3n
qA (n) = n3
s defines the successor function
∗
d defines the doubling function, i.e., d(sn (0)) →
− s2n (0)
2
∗
q defines the square function, i.e., q(sn (0)) →
− sn (0)
from this we get:
∗
2m
m
>22
2m+1
sm := qm+1 (s2 (0)) →
− q(s2 (0)) −−−→ s2
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
(0)
9/21
The Past
Proof of Lemma Á
consider Rhl :
x+0 → x
x+s(y ) → s(x+y )
0A = 2
1
2
3
sA (n) = n + 1
d(0) → 0
d(s(x)) → s(s(d(x)))
q(0) → 0
q(s(x)) → q(x) + s(d(x))
n +A m = n + 2m
dA (n) = 3n
qA (n) = n3
s defines the successor function
∗
d defines the doubling function, i.e., d(sn (0)) →
− s2n (0)
2
∗
q defines the square function, i.e., q(sn (0)) →
− sn (0)
from this we get:
m
2m
∗
2m+1
>22
sm := qm+1 (s2 (0)) →
− q(s2 (0)) −−−→ s2
(0)
we conclude, for all m > 1
m
|sm |−4
dl(sm , →Rhl ) > 22 = 22
where c 6
GM (ICS @ UIBK)
c·|sm |
> 22
1
5
IFIP WG 1.6, July 2, 2009
9/21
The Present
The Present
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
10/21
The Present
How can we improve this ?
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
11/21
The Present
How can we improve this ?
Goal À
modern
study the complexity induced by state-of-the-art termination techniques
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
11/21
The Present
How can we improve this ?
Goal À
modern
study the complexity induced by state-of-the-art termination techniques
Goal Á
useful
induced complexity is bounded by functions of low computational
complexity
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
11/21
The Present
How can we improve this ?
Goal À
modern
study the complexity induced by state-of-the-art termination techniques
. . . basic DP method based on LPO characterises the multiple recursive
functions
Goal Á
useful
induced complexity is bounded by functions of low computational
complexity
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
11/21
The Present
How can we improve this ?
Goal À
modern
study the complexity induced by state-of-the-art termination techniques
. . . basic DP method based on LPO characterises the multiple recursive
functions
Goal Á
useful
induced complexity is bounded by functions of low computational
complexity
. . . WDP method based on POP∗ps induces polytime computability
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
11/21
The Present
How can we improve this ?
Goal À
modern
study the complexity induced by state-of-the-art termination techniques
. . . basic DP method based on LPO characterises the multiple recursive
functions
Goal Á
useful
induced complexity is bounded by functions of low computational
complexity
. . . WDP method based on POP∗ps induces polytime computability
Goal Â
automated
automated complexity analysis
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
11/21
The Present
How can we improve this ?
Goal À
modern
study the complexity induced by state-of-the-art termination techniques
. . . basic DP method based on LPO characterises the multiple recursive
functions
Goal Á
useful
induced complexity is bounded by functions of low computational
complexity
. . . WDP method based on POP∗ps induces polytime computability
Goal Â
automated
automated complexity analysis
X Tyrolean Complexity Tool
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
11/21
The Present
How can we improve this ?
Goal À
modern
study the complexity induced by state-of-the-art termination techniques
. . . basic DP method based on LPO characterises the multiple recursive
functions
Goal Á
useful
induced complexity is bounded by functions of low computational
complexity
. . . WDP method based on POP∗ps induces polytime computability
Goal Â
automated
automated complexity analysis
X Tyrolean Complexity Tool
X Complexity And Termination
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
11/21
The Present
Automated Complexity Analysis: A Snapshot
polynomial derivation length on TPDB
The Present
Automated Complexity Analysis: A Snapshot
polynomial derivation length on TPDB
GM (ICS @ UIBK)
dc–full
(446)
dc–innermost
(1404)
rc–full
(1404)
rc–innermost
(1404)
IFIP WG 1.6, July 2, 2009
12/21
The Present
Automated Complexity Analysis: A Snapshot
polynomial derivation length on TPDB
GM (ICS @ UIBK)
dc–full
(446)
dc–innermost
(1404)
rc–full
(1404)
rc–innermost
(1404)
IFIP WG 1.6, July 2, 2009
POP (40)
fast
12/21
The Present
Automated Complexity Analysis: A Snapshot
polynomial derivation length on TPDB
GM (ICS @ UIBK)
dc–full
(446)
dc–innermost
(1404)
rc–full
(1404)
rc–innermost
(1404)
IFIP WG 1.6, July 2, 2009
POP∗ps (73)
fast
12/21
The Present
Automated Complexity Analysis: A Snapshot
polynomial derivation length on TPDB
GM (ICS @ UIBK)
dc–full
(446)
dc–innermost
(1404)
rc–full
(1404)
rc–innermost
(1404)
IFIP WG 1.6, July 2, 2009
WIDP (214)
still fast
12/21
The Present
Automated Complexity Analysis: A Snapshot
polynomial derivation length on TPDB
GM (ICS @ UIBK)
dc–full
(446)
dc–innermost
(1404)
rc–full
(1404)
rc–innermost
(1404)
IFIP WG 1.6, July 2, 2009
TCT (304)
12/21
The Present
Automated Complexity Analysis: A Snapshot
polynomial derivation length on TPDB
TCT (225)
GM (ICS @ UIBK)
dc–full
(446)
dc–innermost
(1404)
rc–full
(1404)
rc–innermost
(1404)
IFIP WG 1.6, July 2, 2009
TCT (304)
12/21
The Present
Automated Complexity Analysis: A Snapshot
polynomial derivation length on TPDB
CaT (236)
GM (ICS @ UIBK)
dc–full
(446)
dc–innermost
(1404)
rc–full
(1404)
rc–innermost
(1404)
IFIP WG 1.6, July 2, 2009
TCT (304)
12/21
The Present
Automated Complexity Analysis: A Snapshot
polynomial derivation length on TPDB
CaT (236)
TCT (271)
GM (ICS @ UIBK)
CaT (236)
dc–full
(446)
dc–innermost
(1404)
rc–full
(1404)
rc–innermost
(1404)
IFIP WG 1.6, July 2, 2009
TCT (304)
12/21
The Present
Complexity Analysis Based on DP Method
consider Rdiv
1:
2:
GM (ICS @ UIBK)
x −0→x
3:
s(x) − s(y ) → x − y
4:
0 ÷ s(y ) → 0
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
IFIP WG 1.6, July 2, 2009
13/21
The Present
Complexity Analysis Based on DP Method
consider Rdiv
x −0→x
1:
2:
0 ÷ s(y ) → 0
3:
s(x) − s(y ) → x − y
4:
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
Question
what is the runtime complexity of Rdiv ?
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
13/21
The Present
Complexity Analysis Based on DP Method
consider Rdiv
x −0→x
1:
2:
0 ÷ s(y ) → 0
3:
s(x) − s(y ) → x − y
4:
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
Question
what is the runtime complexity of Rdiv ?
Answer
linear
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
13/21
The Present
Complexity Analysis Based on DP Method
consider Rdiv
x −0→x
1:
2:
3:
s(x) − s(y ) → x − y
4:
0 ÷ s(y ) → 0
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
Question
what is the derivational complexity of Rdiv ?
Answer
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
13/21
The Present
Complexity Analysis Based on DP Method
consider Rdiv
x −0→x
1:
2:
3:
s(x) − s(y ) → x − y
4:
0 ÷ s(y ) → 0
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
Question
what is the derivational complexity of Rdiv ?
Answer
at least exponential
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
13/21
The Present
Complexity Analysis Based on DP Method
consider Rdiv
x −0→x
1:
2:
3:
s(x) − s(y ) → x − y
4:
0 ÷ s(y ) → 0
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
Question
what is the innermost runtime complexity of Rdiv ?
Answer
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
13/21
The Present
Complexity Analysis Based on DP Method
consider Rdiv
x −0→x
1:
2:
3:
s(x) − s(y ) → x − y
4:
0 ÷ s(y ) → 0
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
Question
what is the innermost runtime complexity of Rdiv ?
Answer
linear
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
13/21
The Present
Complexity Analysis Based on DP Method
consider Rdiv
x −0→x
1:
2:
3:
s(x) − s(y ) → x − y
4:
0 ÷ s(y ) → 0
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
Question
what is the innermost runtime complexity of Rdiv ?
Answer
linear
Challenge
how to prove (at least) innermost polynomial runtime complexity
automatically?
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
13/21
The Present
Weak Dependency Pairs
Definition
weak dependency pairs
WDP(R) = { l ] → com(u1 ] , . . . , un ] ) | (l → r ) ∈ R, r = C [u1 , . . . , un ]}
|
{z
}
no defined symbols, no vars in C
ui ∈ V or starts with defined functions symbols
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
14/21
The Present
Weak Dependency Pairs
Definition
weak dependency pairs
WDP(R) = { l ] → com(u1 ] , . . . , un ] ) | (l → r ) ∈ R, r = C [u1 , . . . , un ]}
|
{z
}
no defined symbols, no vars in C
ui ∈ V or starts with defined functions symbols; com(t1 , . . . , tn ) is t1 if
n = 1, and c(t1 , . . . , tn ) otherwise
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
14/21
The Present
Weak Dependency Pairs
Definition
weak dependency pairs
WDP(R) = { l ] → com(u1 ] , . . . , un ] ) | (l → r ) ∈ R, r = C [u1 , . . . , un ]}
|
{z
}
no defined symbols, no vars in C
ui ∈ V or starts with defined functions symbols; com(t1 , . . . , tn ) is t1 if
n = 1, and c(t1 , . . . , tn ) otherwise
consider WDP(Rdiv ):
5:
x −] 0 → x
7:
6:
s(x) −] s(y ) → x −] y
8:
GM (ICS @ UIBK)
0 ÷] s(y ) → c
s(x) ÷] s(y ) → (x − y ) ÷] s(y )
IFIP WG 1.6, July 2, 2009
14/21
The Present
Weak Dependency Pairs
Definition
weak dependency pairs
WDP(R) = { l ] → com(u1 ] , . . . , un ] ) | (l → r ) ∈ R, r = C [u1 , . . . , un ]}
|
{z
}
no defined symbols, no vars in C
ui ∈ V or starts with defined functions symbols; com(t1 , . . . , tn ) is t1 if
n = 1, and c(t1 , . . . , tn ) otherwise
consider WDP(Rdiv ):
5:
x −] 0 → x
7:
6:
s(x) −] s(y ) → x −] y
8:
0 ÷] s(y ) → c
s(x) ÷] s(y ) → (x − y ) ÷] s(y )
consider the TRS R: f(s(x)) → g(f(x), f(x))
1 set tn = f(sn (0)), i.e, dl(tn+1 , →R ) > 2n
]
2 but dl(tn+1 , →R∪DP(R) ) = n
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
14/21
The Present
TRS Rdiv :
x −0→x
1:
2:
3:
s(x) − s(y ) → x − y
4:
0 ÷ s(y ) → 0
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
P := WDP(Rdiv ):
5:
x −] 0 → x
7:
6:
s(x) −] s(y ) → x −] y
8:
GM (ICS @ UIBK)
0 ÷] s(y ) → c
s(x) ÷] s(y ) → (x − y ) ÷] s(y )
IFIP WG 1.6, July 2, 2009
15/21
The Present
TRS Rdiv :
x −0→x
1:
2:
3:
s(x) − s(y ) → x − y
4:
0 ÷ s(y ) → 0
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
P := WDP(Rdiv ):
5:
x −] 0 → x
7:
6:
s(x) −] s(y ) → x −] y
8:
0 ÷] s(y ) → c
s(x) ÷] s(y ) → (x − y ) ÷] s(y )
s(0) ÷ s(0) →Rdiv s((0 − 0) ÷ s(0)) →Rdiv s(0 ÷ s(0)) →Rdiv s(0)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
15/21
The Present
TRS Rdiv :
x −0→x
1:
2:
s(x) − s(y ) → x − y
P := WDP(Rdiv ):
5:
x −] 0 → x
7:
6:
s(x) −] s(y ) → x −] y
8:
s(0)÷] s(0) → U (P)∪P
GM (ICS @ UIBK)
0 ÷] s(y ) → c
s(x) ÷] s(y ) → (x − y ) ÷] s(y )
(0−0)÷] s(0) → U (P)∪P
IFIP WG 1.6, July 2, 2009
0÷] s(0) → U (P)∪P
c
15/21
The Present
TRS Rdiv :
x −0→x
1:
2:
s(x) − s(y ) → x − y
P := WDP(Rdiv ):
5:
x −] 0 → x
7:
6:
s(x) −] s(y ) → x −] y
8:
0 ÷] s(y ) → c
s(x) ÷] s(y ) → (x − y ) ÷] s(y )
Lemma
dl(t, →R ) = dl(t ] , →U (WDP(R))∪WDP(R) )
s(0)÷] s(0) → U (P)∪P
GM (ICS @ UIBK)
(0−0)÷] s(0) → U (P)∪P
IFIP WG 1.6, July 2, 2009
0÷] s(0) → U (P)∪P
c
15/21
The Present
TRS Rdiv :
x −0→x
1:
2:
s(x) − s(y ) → x − y
P := WDP(Rdiv ):
5:
x −] 0 → x
7:
6:
s(x) −] s(y ) → x −] y
8:
0 ÷] s(y ) → c
s(x) ÷] s(y ) → (x − y ) ÷] s(y )
Lemma
dl(t, →R ) = dl(t ] , →U (WDP(R))∪WDP(R) )
s(0)÷] s(0) → U (P)∪P
GM (ICS @ UIBK)
(0−0)÷] s(0) → U (P)∪P
IFIP WG 1.6, July 2, 2009
0÷] s(0) → U (P)∪P
c
15/21
The Present
TRS Rdiv :
x −0→x
1:
2:
s(x) − s(y ) → x − y
P := WDP(Rdiv ):
5:
x −] 0 → x
7:
6:
s(x) −] s(y ) → x −] y
8:
0 ÷] s(y ) → c
s(x) ÷] s(y ) → (x − y ) ÷] s(y )
Lemma
dl(t, →R ) = dl(t ] , →U (WDP(R))∪WDP(R) )
s(0)÷] s(0) → U (P)∪P
(0−0)÷] s(0) → U (P)∪P
Definition
0÷] s(0) → U (P)∪P
Fernández
UA(f , R) collects the argument positions of f that are used
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
c
2005
15/21
The Present
Theorem
Hirokawa-M
2008
∀ TRS R, assume
• ∃ rewrite order that induces linear runtime complexity
• U(W(I)DP(R)) ∪ W(I)DP(R) ⊆ then the ( innermost ) runtime complexity of R is linear
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
16/21
The Present
Theorem
Hirokawa-M
2009
∀ TRS R, assume
• ∃ rewrite order that induces linear runtime complexity
• U( WIDP(R)) ∪ WIDP(R) ⊆ and
then the
GM (ICS @ UIBK)
innermost
runtime complexity of R is linear
IFIP WG 1.6, July 2, 2009
16/21
The Present
Theorem
Hirokawa-M
2009
∀ TRS R, assume
• ∃ stable order that induces linear runtime complexity
• U( WIDP(R)) ∪ WIDP(R) ⊆ and
• is monotone on UA(f , R) for any f ∈ F ]
then the
GM (ICS @ UIBK)
innermost
runtime complexity of R is linear
IFIP WG 1.6, July 2, 2009
16/21
The Present
Theorem
Hirokawa-M
2009
∀ TRS R, assume
• ∃ stable order that induces linear runtime complexity
• U( WIDP(R)) ∪ WIDP(R) ⊆ and
• is monotone on UA(f , R) for any f ∈ F ]
• WIDP(R) is constructor
then the
GM (ICS @ UIBK)
innermost
runtime complexity of R is linear
IFIP WG 1.6, July 2, 2009
16/21
The Present
Theorem
Hirokawa-M
2009
∀ TRS R, assume
• ∃ stable order that induces linear runtime complexity
• U( WIDP(R)) ∪ WIDP(R) ⊆ and
• is monotone on UA(f , R) for any f ∈ F ]
• WIDP(R) is constructor
then the
innermost
runtime complexity of R is linear
consider
U(P) ∪ P and the WMA A:
0A = cA = 0
sA (x) = x + 2
−A (x, y ) = −]A (x, y ) = ÷]A (x, y ) = x + 1
1
A is monotone on usable arguments
2
P ⊆ >A and U(P) ⊆ >A
we conclude linear innermost runtime complexity
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
16/21
The Future
The Future
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
17/21
The Future
Open Problems and Challenges
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
18/21
The Future
Open Problems and Challenges
Goal À
modern
. . . complexity-wise it is the removal of rules that adds real power to the
DP method
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
18/21
The Future
Open Problems and Challenges
Goal À
modern
. . . complexity-wise it is the removal of rules that adds real power to the
DP method
Goal Á
useful
. . . how to remove all the extra conditions from the WDP method
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
18/21
The Future
Open Problems and Challenges
Goal À
modern
. . . complexity-wise it is the removal of rules that adds real power to the
DP method
Goal Á
useful
. . . how to remove all the extra conditions from the WDP method
Goal Ã
applications
program analysis, completion, automated deduction, implicit
computational complexity theory, proof theory . . .
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
18/21
The Future
Open Problems and Challenges
Goal À
modern
. . . complexity-wise it is the removal of rules that adds real power to the
DP method
Goal Á
useful
. . . how to remove all the extra conditions from the WDP method
Goal Ã
applications
program analysis, completion, automated deduction, implicit
computational complexity theory, proof theory . . .
. . . complexity preserving transformations from Scheme, Haskell, Logic
Programs, JAVA bytecode . . .
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
18/21
The Future
Thank you for Your Attention!
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
19/21
Complexity in Rewriting: A History
number of complexity related papers
RTA papers
(539)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
20/21
Complexity in Rewriting: A History
number of complexity related papers
keyword “complexity” (30)
RTA papers
(539)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
20/21
Complexity in Rewriting: A History
number of complexity related papers
complexity of TRSs (15)
keyword “complexity” (30)
RTA papers
(539)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
20/21
Complexity in Rewriting: A History
number of complexity related papers
termination (98)
complexity of TRSs (15)
keyword “complexity” (30)
RTA papers
(539)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
20/21
Complexity in Rewriting: A History
number of complexity related papers
TCS & IC
(942)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
20/21
Complexity in Rewriting: A History
number of complexity related papers
keyword “complexity” (58)
TCS & IC
(942)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
20/21
Complexity in Rewriting: A History
number of complexity related papers
complexity of TRSs (5)
keyword “complexity” (58)
TCS & IC
(942)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
20/21
Complexity in Rewriting: A History
number of complexity related papers
termination (89)
complexity of TRSs (5)
keyword “complexity” (58)
TCS & IC
(942)
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
20/21
Usable Argument Positions
Definition
let R, P be a pair of TRSs based on signature F, such that P is a
constructur TRS with respect to F. The set of usable arguments of f ∈ F
with respect to R, P is defined as follows.
UA(f , R, P) := {1 6 i 6 n | ∃l → r ∈ R∪P, ∃p, p 0 ∈ Pos(r ) such}
that p 0 .i 6 p, root(r |p ) is defined in
R, root(s|p0 ) = f , and l 6B r|p
Here the arity of f is n.
GM (ICS @ UIBK)
IFIP WG 1.6, July 2, 2009
21/21
Usable Argument Positions
Definition
let R, P be a pair of TRSs based on signature F, such that P is a
constructur TRS with respect to F. The set of usable arguments of f ∈ F
with respect to R, P is defined as follows.
UA(f , R, P) := {1 6 i 6 n | ∃l → r ∈ R∪P, ∃p, p 0 ∈ Pos(r ) such}
that p 0 .i 6 p, root(r |p ) is defined in
R, root(s|p0 ) = f , and l 6B r|p
Here the arity of f is n.
x −0→x
1:
s(x) − s(y ) → x − y
2:
0 ÷ s(y ) → 0
3:
4:
s(x) ÷ s(y ) → s((x − y ) ÷ s(y ))
usable argument positions are as follows
UAR (0) = UAR (−) = ∅
GM (ICS @ UIBK)
UAR (s) = UAR (÷) = {1}
IFIP WG 1.6, July 2, 2009
21/21